Haitian Creole is derived from Latin, although the Haitians generally have very little Roman ancestry, I would guess. Languages have often been adopted by people whose ancestors spoke different languages (a great many Americans are in that category). It would not be very surprising if the Indo-European languages spread in India without a large influx of new people, if the new people happened to be in control
The word "punch" and Punjab (Panjab) are both derived from the Sanskrit word for "five" (Punjab is a region with five rivers, and punch had five ingredients), which is similar to the Greek word pente meaning "five."
English "five" is also related, but the original p changed to f in the Germanic languages (Greek pater, English "father," Latin piscis, English "fish," Greek polos, English "foal," etc.)
When one major culture adopts the tongue of another, they usually do so because the adopted language has a specific advantage, most always the advantage being of the language having a written script(the Thai script is a stylised Indian Tamil script). But Indian languages, given the similarites with ancient Greek, have a totally different script compared to the European languages.
So this theory that Indians borrowed only language, of the spoken form, and not the script, has some holes to patch.